Heat the butter till it bubbles in a large stainless-steel skillet. Add the green onions and garlic and cook until the garlic is fragrant.
Add the Cognac to the pan. Warm it and either boil it off or flame it. (Very carefully.)
Add the Creole seasoning, salt, tarragon, and cream. Bring the pan to a boil while agitating the contents carefully to blend. Reduce the cream by about a third (about three minutes over medium-high heat). Then add the crawfish tails and cook until heated through.
Add the cooked, drained pasta to the pan and toss with the sauce to distribute all the ingredients and sauce uniformly. Serve immediately, garnished with finely-chopped green onions. (Resist the temptation to add Parmesan or Romano cheese.)